# One-form

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In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.

In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to R whose restriction to each fibre is a linear functional on the tangent space. Symbolically,

${\displaystyle \alpha :TM\rightarrow {\mathbf {R} },\quad \alpha _{x}=\alpha |_{T_{x}M}:T_{x}M\rightarrow {\mathbf {R} }}$

where αx is linear.

Often one-forms are described locally, particularly in local coordinates. In a local coordinate system, a one-form is a linear combination of the differentials of the coordinates:

${\displaystyle \alpha _{x}=f_{1}(x)\,dx_{1}+f_{2}(x)\,dx_{2}+\cdots +f_{n}(x)\,dx_{n}}$

where the fi are smooth functions. From this perspective, a one-form has a covariant transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covariant tensor field.

## Examples

Many real-world concepts can be described as one-forms:

• Indexing into a vector: The second element of a three-vector is given by the one-form [0, 1, 0]. That is, the second element of [x ,y ,z] is
[0, 1, 0] · [xyz] = y.
• Mean: The mean element of an n-vector is given by the one-form [1/n, 1/n, ..., 1/n]. That is,
${\displaystyle \operatorname {mean} (v)=[1/n,1/n,\dots ,1/n]\cdot v.}$
• Sampling: Sampling with a kernel can be considered a one-form, where the one-form is the kernel shifted to the appropriate location.
${\displaystyle \mathrm {NPV} (R(t))=\langle w,R\rangle =\int _{t=0}^{\infty }{\frac {R(t)}{(1+i)^{t}}}\,dt.}$

## Differential of a function

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Let ${\displaystyle U\subseteq \mathbb {R} }$ be open (e.g., an interval ${\displaystyle (a,b)}$), and consider a differentiable function ${\displaystyle f:U\to \mathbb {R} }$, with derivative f'. The differential df of f, at a point ${\displaystyle x_{0}\in U}$, is defined as a certain linear map of the variable dx. Specifically, ${\displaystyle df(x_{0},dx):dx\mapsto f'(x_{0})dx}$. (The meaning of the symbol dx is thus revealed: it is simply an argument, or independent variable, of the function df.) Hence the map ${\displaystyle x\mapsto df(x,dx)}$ sends each point x to a linear functional df(x,dx). This is the simplest example of a differential (one-)form.

In terms of the de Rham complex, one has an assignment from zero-forms (scalar functions) to one-forms i.e., ${\displaystyle f\mapsto df}$.